The Szilárd engine is the simplest realisation of the Maxwell's
demon idea, that the second law of thermodynamics can be violated if
some information on the microscopic state of a system is available. In
its original form, it consists of a box of unit lateral length in each
direction, containing a single point-like particle. The box is in
contact with a heat bath at temperature \(T\) . At a certain time \(t\) , a
measurement \(Y\)  is taken that reveals whether the particle is in the
left (\(Y=0\) ) or the right (\(Y=1\) ) side of the box. This allows the
observer to extract work from this system, using the following
protocol. A wall is inserted without friction and instantaneously
right after the measurement. The measurement reveals on which side the
particle is confined. Hence the observer knows on which side the
particle will push the wall. The force exerted on the wall by the
particle can be exploited to do work, e.g. to lift a weight. There are
several details in this idealised system that have been discussed at length
elsewhere [1]}, [2]}, [3]}. In
the present context, let it suffice to say that in the limit of a
quasi-static process (i.e. when the wall moves infinitely slowly), the
work that can be extracted can be computed considering the expansion
of an ideal gas composed of a single particle. This is characterised
by an equation of state \(PV=Nk_BT\) , where \(N\)  is the number of
particles and \(k_B\)  is Boltzmann constant, that we shall take equal to
one (\(k_B=1\) ) in what follows. After the wall is inserted, the
pressure on the side containing the particle is \(P=T/V\) , since
\(N=1\) . On the other side the pressure is zero, because \(N=0\) . In an
isothermal expansion, the work done for a change \(dV\)  in the volume is
\(dW=PdV\) . Integrating this from the initial volume \(V_i=1/2\)  to the
final volume \(V_f=1\)  yields,
\(W\;=\;\int _{1/2}^1\frac{dV}{V}\;=\;T\log 2.\) 
